LipschitzianQ-matrices areP-matrices

Murthy, G. S. R.; Parthasarathy, T.; Sabatini, Marco

doi:10.1007/bf02592146

Cited by 12 publications

(3 citation statements)

References 4 publications

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“…Gowda [8] proved that if L 0 (q) = {0} for some positive q, then the Lipschitz continuity of L 0 everywhere in the sense of the Hausdorff metric guarantees that L 0 is single-valued everywhere; see also Pang [20]. Murthy, Parthasarathy and Sabatini [18] dropped the requirement that L 0 (q) = {0} for some positive q, obtaining that L 0 is Lipschitz continuous everywhere if and only if it is single-valued everywhere. Gowda and Sznajder [9] noted that this result can be extended to the map L by using some recently discovered properties of normal maps.…”

Section: Introductionmentioning

confidence: 99%

Characterizations of Strong Regularity for Variational Inequalities over Polyhedral Convex Sets

Dontchev¹,

Rockafellar²

1996

SIAM J. Optim.

246

190

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Linear and nonlinear variational inequality problems over a polyhedral convex set are analyzed parametrically. Robinson's notion of strong regularity, as a criterion for the solution set to be a singleton depending Lipschitz continuously on the parameters, is characterized in terms of a new "critical face" condition and in other ways. The consequences for complementarity problems are worked out as a special case. Application is also made to standard nonlinear programming problems with parameters that include the canonical perturbations. In that framework a new characterization of strong regularity is obtained for the variational inequality associated with the Karush-Kuhn-Tucker conditions.

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Section: Introductionmentioning

confidence: 99%

Characterizations of Strong Regularity for Variational Inequalities over Polyhedral Convex Sets

Dontchev¹,

Rockafellar²

1996

SIAM J. Optim.

246

190

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“…Gowda [6] showed that if A is Lipschitzian and (q, A) has a unique solution for some nondegenerate q (q is said to be nondegenerate if z + Az + q > 0 for all z ∈ S(q, A)), then A is in P . Later, Murthy, Parthasarathy, and Sabatini [9] showed that a Q-matrix is Lipschitzian if it is a P -matrix. Stone [15] showed that Lipschitzian matrices are nondegenerate INS-matrices and conjectured that the converse is also true.…”

Section: Results On P 1 -Matricesmentioning

confidence: 99%

A Note on P₁ - and Lipschitzian Matrices

Murthy¹,

Neogy²,

Thuijsman³

2000

SIAM J. Matrix Anal. & Appl.

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The linear complementarity problem (q, A) with data A ∈ R n×n and q ∈ R n involves finding a nonnegative z ∈ R n such that Az + q ≥ 0 and z t (Az + q) = 0. Cottle and Stone introduced the class of P 1-matrices and showed that if A is in P 1 \Q, then K(A) (the set of all q for which (q, A) has a solution) is a half-space and (q, A) has a unique solution for every q in the interior of K(A). Extending the results of Murthy, Parthasarathy, and Sriparna [Ann. Dynamic Games, to appear], we present a number of equivalent characterizations of P 1 \Q. Also, we present yet another characterization of P-matrices. This widens the range of matrix classes for which a conjecture raised by Murthy, Parthasarathy, and Sriparna [SIAM J. Matrix Anal. Appl., 19 (1998), pp. 898-905] characterizing the class of Lipschitzian matrices is true.

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“…In particular, Mangasarian and Shiau 14 showed that if M is a P -matrix, then solutions of linear inequalities, programs, and LCP are Lipschitz continuous. Murthy et al 15 showed that M is a P -matrix if and only if the LCP M, q has a solution for all q ∈ R n and the solution mapping is Lipschitzian. Gowda and Sznajder 16 generalized the above result to affine variational inequality problems, while Yen 17 studied Lipschitz continuity of the solution mapping of variational inequalities with a parametric polyhedral constraint.…”

Section: 2mentioning

confidence: 99%

Lipschitz Continuity of the Solution Mapping of Symmetric ConeComplementarity Problems

Miao

Chen

2012

Abstract and Applied Analysis

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This paper investigates the Lipschitz continuity of the solution mapping of symmetric cone (linear or nonlinear) complementarity problems (SCLCP or SCCP, resp.) over Euclidean Jordan algebras. We show that if the transformation has uniform CartesianP-property, then the solution mapping of the SCCP is Lipschitz continuous. Moreover, we establish that the monotonicity of mapping and the Lipschitz continuity of solutions of the SCLCP imply ultraP-property, which is a concept recently developed for linear transformations on Euclidean Jordan algebra. For a Lyapunov transformation, we prove that the strong monotonicity property, the ultraP-property, the CartesianP-property, and the Lipschitz continuity of the solutions are all equivalent to each other.

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LipschitzianQ-matrices areP-matrices

Cited by 12 publications

References 4 publications

Characterizations of Strong Regularity for Variational Inequalities over Polyhedral Convex Sets

Characterizations of Strong Regularity for Variational Inequalities over Polyhedral Convex Sets

A Note on P₁ - and Lipschitzian Matrices

Lipschitz Continuity of the Solution Mapping of Symmetric ConeComplementarity Problems

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LipschitzianQ-matrices areP-matrices

Cited by 12 publications

References 4 publications

Characterizations of Strong Regularity for Variational Inequalities over Polyhedral Convex Sets

Characterizations of Strong Regularity for Variational Inequalities over Polyhedral Convex Sets

A Note on P1 - and Lipschitzian Matrices

Lipschitz Continuity of the Solution Mapping of Symmetric ConeComplementarity Problems

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Product

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A Note on P₁ - and Lipschitzian Matrices